In the board game Monopoly, players roll two fair six-sided dice on each of their turns. The sum of the two dice determines how many spaces the player moves his or her game piece on the board. If the two dice roll the same number, this is called “doubles.” Whenever a player rolls doubles, that player rolls again and moves his or her game piece again. However, if a player rolls doubles three times on his or her turn, the player moves directly to "jail" (which we will considered a total move of 0 -- ignoring all previous moves).
What is the expected value of the number of spaces a player moves his or her game piece during that player’s turn?
The solution does this by conditioning on the double rolls; it made sense but my answer is very slightly off and i'm wondering why. Label $E$ as the expected value. Label $E_d$ as the expected value of a double roll. Label $E_{dd}$ as the expected value of two double rolls. $E_{ddd}$ is simply zero. I created the following system of linear equations.
(1) $ E =\frac{5}{6}(7) + \frac{1}{6}(E_{d})$ (2) $E_{d} = \frac{5}{6}(7+7) + \frac{1}{6}(E_{dd})$ (3) $E_{dd} = \frac{5}{6}(7 + E_{d}) + \frac{1}{6}(0)$
Solving i'm off by 0.01. What's the problem?